Jessie Zhao Course page: 1

 For any integer n>0, the number of permutations of a set with n elements is n! ◦ A permutation of a set of elements is an ordering of the elements. ◦ E.g. the set of elements {a, b, c} can be ordered in the following ways:  abc acb cba bac bca cab ◦ By the product rule, there are n(n-1)(n-2)..1=n! permutations 2

 Ex: Suppose there are 50 students in the class, ◦ In how many ways can the whole class stand in a line? 50! ◦ In how many ways can we select three students to stand in a line? 50*49*48 3

 An r-permutation is an ordering of r elements of a set of n elements, denoted by P(n,r) ◦ E.g. the 2-permutations of the set of elements {a, b, c} are:  ab ac ba bc ca cb ◦ By the product rule, there are n(n-1)(n-2)....(n-r+1) r-permutations 4

 P(n,r) = n!/(n-r)! = n(n-1)(n-2)....(n-r+1) for 0≤r≤n  Special cases: ◦ P(n,0) = 1 ◦ P(0,0) = 1 ◦ P(n,n) = n! 5

 Recall: How many one-to-one functions are there from a set with m elements to one with n elements? ◦ n(n-1)...(n-m+1) when m≤n ◦ 0 when m>n 6

 For the solitaire hand that show initially ◦ How many possible hands? p(52,7) ◦ How many possible hands with no Aces? p(48,7) ◦ How many possible hands with one or more Aces? P(52,7)-P(48,7) 7

 An r-combination is an unordered selection of r elements of a set of n elements, denoted by C(n,r)  E.g. the 2-combinations of the set of elements {a,b, c} are: ◦ {a,b} {a,c} {b,c} 8

 There are r! permutation of each subset  There are more r-permutation than r- combinations. 9

10 Corollary: C(n,r) = C(n,n-r)

 For a deck of 52 cards,  How many poker hands of five cards can there be? C(52,5)=2,598,960  How many ways are there to select 47 cards? C(52,47)=C(52,5)=2,598,960 11